A101925 - OEIS

A101925

a(n) = A005187(n) + 1.

1, 2, 4, 5, 8, 9, 11, 12, 16, 17, 19, 20, 23, 24, 26, 27, 32, 33, 35, 36, 39, 40, 42, 43, 47, 48, 50, 51, 54, 55, 57, 58, 64, 65, 67, 68, 71, 72, 74, 75, 79, 80, 82, 83, 86, 87, 89, 90, 95, 96, 98, 99, 102, 103, 105, 106, 110, 111, 113, 114, 117, 118, 120, 121, 128, 129

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Exponent of 2 in the sequences A032184, A052278, A060055, A066318, A088229, A101926.

p(n) sequence for k=2, s=0. p(n) = min(j: A046699(j) = n). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

LINKS

Table of n, a(n) for n=0..65.

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences

B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.

F. Ruskey, C. Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]

FORMULA

Recurrence: a(2n) = 2a(n) + A000120(n) - 1, a(2n+1) = a(2n) + 1.

G.f.: (1 / 1-z) * (z + z * sum(z^(2^i) * (s + (1 / (1 - z^(2^k)))),i=0..infinity)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

MATHEMATICA

Table[IntegerExponent[(2 n)!, 2] + 1, {n, 0, 65}] (* or *)

Table[2 n - DigitCount[2 n, 2, 1] + 1, {n, 0, 65}] (* Michael De Vlieger, Feb 04 2017 *)

PROG

(PARI) a(n)=1+sum(k=1, n, valuation(k, 2)+1)

(PARI) a(n)=if(n==0, 1, if((n%2)==0, 2*a(n/2)+subst(Pol(binary(n)), x, 1)-1, a(n-1)+1))

(PARI) a(n)=2*n+1-hammingweight(n) \\ Charles R Greathouse IV, Dec 29 2022

(Python 3.10+)

def A101925(n): return (n<<1)-n.bit_count()+1 # Chai Wah Wu, Jul 13 2022